Bulletin of the Australian Mathematical Society

Table of Contents - June 2012 - Volume 85, Issue 03  

Research Article

ON SOME REFINEMENTS AND CONVERSES OF MULTIDIMENSIONAL HILBERT-TYPE INEQUALITIES

MARIO KRNIĆa1

a1 Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia (email: mario.krnic@fer.hr)

Abstract

The main objective of this paper is a study of some new refinements and converses of multidimensional Hilbert-type inequalities with nonconjugate exponents. Such extensions are deduced with the help of some remarkable improvements of the well-known Hölder inequality. First, we obtain refinements and converses of the general multidimensional Hilbert-type inequality in both quotient and difference form. We then apply the results to homogeneous kernels with negative degree of homogeneity. Finally, we consider some particular settings with homogeneous kernels and weighted functions, and compare our results with those in the literature.

(Received June 03 2011)

2010 Mathematics subject classification

  • primary 26D15

Keywords and phrases

  • Hilbert-type inequality;
  • nonconjugate exponents;
  • Hölder’s inequality;
  • refinement;
  • converse

Footnotes

This research was supported by the Croatian Ministry of Science, Education, and Sports, under Research Grant 036–1170889–1054.